#### Presentation Title

The Topological Structure of the Biquaternionic Projective Point

#### Start Date

12-11-2016 11:00 AM

#### End Date

12-11-2016 11:15 AM

#### Location

HUB 355

#### Type of Presentation

Oral Talk

#### Abstract

The biquaternions **B** are a noncommutative algebra that contains zero divisors. It has been previously shown that a quotienting process can be employed to construct the biquaternionic projective point **BP ^{0}** from

**B**. It is known that

**BP**possesses a natural twistor structure that comes about from properties within its topology.

^{0}^{1}While it is understood that the twistor structure arises from the quotient topology of

**BP**, the topology itself is not well understood. In this paper we seek to explore the topological structure of

^{0}**BP**. We will show that

^{0}**BP**has an exotic topology that possesses many interesting properties due to the presence of a dense point in its structure. Here, we will begin by characterizing a basis for the topology which we will utilize to show that the dense point lies in every nonempty open subset of

^{0}**BP**. We then use this fact to show that

^{0}**BP**is connected. Finally, we construct derived sets for arbitrary open and closed subsets of

^{0}**BP**, and show that every nonempty open subset is dense and every proper closed subset has an empty interior.

^{0}1. Agnew, A. F. (2003). The Twistor Structure of the Biquaternionic Projective Point. Advances in Applied Clifford Algebras, 13(2), 231-240.

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The Topological Structure of the Biquaternionic Projective Point

HUB 355

The biquaternions **B** are a noncommutative algebra that contains zero divisors. It has been previously shown that a quotienting process can be employed to construct the biquaternionic projective point **BP ^{0}** from

**B**. It is known that

**BP**possesses a natural twistor structure that comes about from properties within its topology.

^{0}^{1}While it is understood that the twistor structure arises from the quotient topology of

**BP**, the topology itself is not well understood. In this paper we seek to explore the topological structure of

^{0}**BP**. We will show that

^{0}**BP**has an exotic topology that possesses many interesting properties due to the presence of a dense point in its structure. Here, we will begin by characterizing a basis for the topology which we will utilize to show that the dense point lies in every nonempty open subset of

^{0}**BP**. We then use this fact to show that

^{0}**BP**is connected. Finally, we construct derived sets for arbitrary open and closed subsets of

^{0}**BP**, and show that every nonempty open subset is dense and every proper closed subset has an empty interior.

^{0}1. Agnew, A. F. (2003). The Twistor Structure of the Biquaternionic Projective Point. Advances in Applied Clifford Algebras, 13(2), 231-240.